1. Field of the Invention
This invention relates broadly to statistical analysis of random variables that simulate a property field. More particularly, this invention relates to multipoint geostatistical methods and simulations.
2. State of the Art
Geostatistics is a discipline concerned with spatially distributed random variables (also called “regionalized variables”), usually applied to problems in the earth sciences, such as estimation of mineral reserves and delineation of mineral deposits, hydrocarbon reservoirs, and aquifers. Multipoint (or multiple-point) geostatistics (MPGS) differs from the rest of geostatistics primarily in that it characterizes spatial variability using patterns (sets of points) that contain more than two points.
One of the goals of multipoint geostatistics is simulation, namely the generation of numerical values along a line, on a surface, or in a volume, such that the set of values honors certain given spatial correlation properties (usually derived from a data set called “analog” or “training image”) while optionally (in the case called “conditional simulation”) honoring predetermined data. In practice, the “analog” may be, for example, a well-known rock volume that is statistically similar to a yet poorly known oil reservoir being delineated, and the predetermined data to be honored may be lithology observations at wells, or probabilities of lithologies derived from seismic data. In this manner, MPGS simulations honor absolute or so-called “hard” constraints from data acquired in wells or outcrops, and conditional or “soft” constraints from seismic data, facies probability fields, and rotation and affinity (or scale) constraint grids. Such data are used in a stochastic modeling process to generate one-dimensional (1D), two-dimensional (2D) and/or three-dimensional (3D) maps of geological facies or rock properties. Since there is a random component involved in MPGS simulations, individual realizations of property fields created by MPGS algorithms differ, but the ensemble of realizations provide geoscientists and reservoir engineers with improved quantitative estimates of the spatial distribution and uncertainty of geological facies or rock properties in a modeled reservoir volume.
Multipoint geostatistical methods have been recently demonstrated to be computationally feasible and have been tested on real datasets as set forth in i) Strebelle, “Conditional Simulation of complex geological structures using multiple-point statistics,” Mathematical Geology, v. 34, n. 1, 2002, pp. 1-22, ii) Strebelle et al., “Modeling of a deepwater turbidite reservoir conditional to seismic data using principal component analysis and multiple-point geostatistics,” SPE Journal, Vol. 8, No. 3, 2003, pp. 227-235, and iii) Liu et al., “Multiple-point simulation integrating wells, three-dimensional seismic data, and geology,” American Association of Petroleum Geologists Bulletin v.88, no. 7, 2004, pp. 905-921.
Traditional geostatistical methods rely upon a variogram to describe geologic continuity. However, a variogram, which is a two-point measure of spatial variability, cannot describe realistic, curvilinear or geometrically complex patterns. Multipoint geostatistical methods use a training image (instead of a variogram) to account for geological information. The training image provides a conceptual description of the subsurface geological heterogeneity, containing possibly complex multipoint patterns of geological heterogeneity. Multipoint statistics simulation anchors these patterns to well data (and/or outcrop data) and to the seismic-derived information (and/or probability field information or constraint grid(s)).
The training image often, but not necessarily, has a low-resolution (i.e. typically a few hundred pixels a side). Each pixel of the training image has a level (which can be a binary value, a grayscale value or a color value) associated therewith. The level at each pixel is referred to herein as a category. Typically, there are around 5 to 10 possible categories at each pixel of the training image, but this number could be greater or smaller. The shapes of geological element(s) defined by the training image represent a model of real geological elements, with each category typically representing a different geological facies or a different kind of geological body.
Geostatistics relies on the well-known concept of random variables. In simple terms, reservoir properties at various grid locations are largely unknown or uncertain; hence each property of interest at each grid location is turned into a random variable whose variability is described by a probability function. In order to perform any type of geostatistical simulation, one requires a decision or assumption of stationarity. In multipoint geostatistical methods, the use of training images is bound by the principle of stationarity as described by Caers et al., “Multiple-point geostatistics: a quantitative vehicle for integrating geologic analogs into multiple reservoir models,” AAPG Memoir, Integration of Outcrop and Modern Analogs in Reservoir Modeling, (eds.) Grammer, G. M, 2002. In the case of 2D or 3D reservoir modeling, a random variable or random process is said to be stationary if all of its statistical parameters are independent of its location in space (invariant according to any translation). In the case of training images, this stationarity can consist, but is not limited to,                orientation stationarity, where directional elements do not rotate across the training image; and        scale stationarity (where the size of elements on the image does not change across the training image).        
Although the concept of stationarity is referred to in Caers et al. paper entitled “Multiple-point geostatistics: a quantitative vehicle for integrating geologic analogs into multiple reservoir models,” this paper fails to disclose a methodology for automatically estimating and validating the stationarity of a given training image to ensure that that training image is appropriate for multipoint geostatistical methods.